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Conjugacy growth and hyperbolicity2. Conjugacy growth series in groups
3. Rivin’s conjecture for hyperbolic groups
4. Conjugacy representatives in acylindrically hyperbolic groups
2 / 48
Let G be a group with finite generating set X .
I Denote by [g ] the conjugacy class of g ∈ G
and by |g |c the conjugacy
length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect
to X .
σG ,X (n) := ]{[g ] ∈ G | |g |c = n}.
3 / 48
Let G be a group with finite generating set X .
I Denote by [g ] the conjugacy class of g ∈ G and by |g |c the conjugacy
length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect
to X .
σG ,X (n) := ]{[g ] ∈ G | |g |c = n}.
3 / 48
Let G be a group with finite generating set X .
I Denote by [g ] the conjugacy class of g ∈ G and by |g |c the conjugacy
length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect
to X .
σG ,X (n) := ]{[g ] ∈ G | |g |c = n}.
3 / 48
I Guba-Sapir (2010): asymptotics of the conjugacy growth function for
BS(1, n), the Heisenberg group on two generators, diagram groups, some
HNN extensions.
I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type
‘monsters’) groups of standard exponential growth should have
exponential conjugacy growth.
conjugacy growth for f.g. linear (non virt. nilpotent) groups.
I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it
is possible to construct groups with a prescribed conjugacy growth
function.
I Guba-Sapir (2010): asymptotics of the conjugacy growth function for
BS(1, n), the Heisenberg group on two generators, diagram groups, some
HNN extensions.
I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type
‘monsters’) groups of standard exponential growth should have
exponential conjugacy growth.
conjugacy growth for f.g. linear (non virt. nilpotent) groups.
I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it
is possible to construct groups with a prescribed conjugacy growth
function.
I Guba-Sapir (2010): asymptotics of the conjugacy growth function for
BS(1, n), the Heisenberg group on two generators, diagram groups, some
HNN extensions.
I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type
‘monsters’) groups of standard exponential growth should have
exponential conjugacy growth.
conjugacy growth for f.g. linear (non virt. nilpotent) groups.
I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it
is possible to construct groups with a prescribed conjugacy growth
function.
I Guba-Sapir (2010): asymptotics of the conjugacy growth function for
BS(1, n), the Heisenberg group on two generators, diagram groups, some
HNN extensions.
I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type
‘monsters’) groups of standard exponential growth should have
exponential conjugacy growth.
conjugacy growth for f.g. linear (non virt. nilpotent) groups.
I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it
is possible to construct groups with a prescribed conjugacy growth
function.
Conjugacy growth in geometry
A slight modification of the conjugacy growth function (including only the
non-powers) appears in geometry:
- counting the primitive closed geodesics of bounded length on a compact
manifold M of negative curvature and exponential volume growth gives, via
quasi-isometries, good (exponential) asymptotics for σ(n) for the fundamental
group of M (Margulis, . . . ).
Conjugacy growth in geometry
A slight modification of the conjugacy growth function (including only the
non-powers) appears in geometry:
- counting the primitive closed geodesics of bounded length on a compact
manifold M of negative curvature and exponential volume growth gives,
via
group of M (Margulis, . . . ).
Conjugacy growth in geometry
A slight modification of the conjugacy growth function (including only the
non-powers) appears in geometry:
- counting the primitive closed geodesics of bounded length on a compact
manifold M of negative curvature and exponential volume growth gives, via
quasi-isometries, good (exponential) asymptotics for σ(n) for the fundamental
group of M (Margulis, . . . ).
Let G be a group with finite generating set X .
I The conjugacy growth series of G with respect to X records the number of
conjugacy classes of every length. It is
σ(G ,X )(z) := ∞∑ n=0
σ(G ,X )(n)zn,
where σ(G ,X )(n) is the number of conjugacy classes of length n.
6 / 48
Conjecture (Rivin, 2000)
If G hyperbolic, then the conjugacy growth series of G is rational if and only if
G is virtually cyclic.
Theorem (Antoln-C., 2015)
If G is non-elementary hyperbolic, then the conjugacy growth series is
transcendental.
Theorem (C., Hermiller, Holt, Rees, 2014)
Let G be a virtually cyclic group. Then the conjugacy growth series of G is
rational.
NB: Both results hold for all symmetric generating sets of G .
7 / 48
Conjecture (Rivin, 2000)
If G hyperbolic, then the conjugacy growth series of G is rational if and only if
G is virtually cyclic.
Theorem (Antoln-C., 2015)
If G is non-elementary hyperbolic, then the conjugacy growth series is
transcendental.
Theorem (C., Hermiller, Holt, Rees, 2014)
Let G be a virtually cyclic group. Then the conjugacy growth series of G is
rational.
NB: Both results hold for all symmetric generating sets of G .
7 / 48
Conjecture (Rivin, 2000)
If G hyperbolic, then the conjugacy growth series of G is rational if and only if
G is virtually cyclic.
Theorem (Antoln-C., 2015)
If G is non-elementary hyperbolic, then the conjugacy growth series is
transcendental.
Theorem (C., Hermiller, Holt, Rees, 2014)
Let G be a virtually cyclic group. Then the conjugacy growth series of G is
rational.
NB: Both results hold for all symmetric generating sets of G . 7 / 48
Conjugacy representatives
In order to determine the conjugacy growth series, we need a set of minimal
length conjugacy representatives,
exactly one word w ∈ X ∗ such that
1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and
2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where
I l(w) := word length of w ∈ X∗
I |g | = |g |X := the (group) length of g ∈ G with respect to X .
8 / 48
Conjugacy representatives
In order to determine the conjugacy growth series, we need a set of minimal
length conjugacy representatives, i.e. for each conjugacy class [g ] in G pick
exactly one word w ∈ X ∗ such that
1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and
2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where
I l(w) := word length of w ∈ X∗
I |g | = |g |X := the (group) length of g ∈ G with respect to X .
8 / 48
Conjugacy representatives
In order to determine the conjugacy growth series, we need a set of minimal
length conjugacy representatives, i.e. for each conjugacy class [g ] in G pick
exactly one word w ∈ X ∗ such that
1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and
2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where
I l(w) := word length of w ∈ X∗
I |g | = |g |X := the (group) length of g ∈ G with respect to X .
8 / 48
Conjugacy growth series in virt. cyclic groups: Z, Z2 ∗ Z2
In Z the conjugacy growth series is the same as the standard one:
σ(Z,{1,−1})(z) = 1 + 2z + 2z2 + · · · = 1 + z
1− z .
In Z2 ∗ Z2 a set of conjugacy representatives is 1, a, b, ab, abab, . . . , so
σ(Z2∗Z2,{a,b})(z) = 1 + 2z + z2 + z4 + z6 · · · = 1 + 2z − 2z3
1− z2 .
9 / 48
Conjugacy growth series in virt. cyclic groups: Z, Z2 ∗ Z2
In Z the conjugacy growth series is the same as the standard one:
σ(Z,{1,−1})(z) = 1 + 2z + 2z2 + · · · = 1 + z
1− z .
In Z2 ∗ Z2 a set of conjugacy representatives is 1, a, b, ab, abab, . . . , so
σ(Z2∗Z2,{a,b})(z) = 1 + 2z + z2 + z4 + z6 · · · = 1 + 2z − 2z3
1− z2 .
9 / 48
Conjugacy growth series in free groups: F2 = a, b
Set a < b < a−1 < b−1 and choose as conjugacy representative the smallest
shortlex rep. in each conjugacy class, so the language is
{a±k , b±k , ab, ab−1, ba−1, a−1b−1, a2b,aba, · · · }
10 / 48
Asymptotics of conjugacy growth in the free group
Idea: take all cyclically reduced words of length n, whose number
is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n.
Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy
growth function is given by
σp(n) ∼ (2k − 1)n+1
, h = log(2k − 1).
In general, when powers are included, one cannot divide by n.
11 / 48
Asymptotics of conjugacy growth in the free group
Idea: take all cyclically reduced words of length n, whose number
is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n.
Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy
growth function is given by
σp(n) ∼ (2k − 1)n+1
, h = log(2k − 1).
In general, when powers are included, one cannot divide by n.
11 / 48
Asymptotics of conjugacy growth in the free group
Idea: take all cyclically reduced words of length n, whose number
is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n.
Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy
growth function is given by
σp(n) ∼ (2k − 1)n+1
, h = log(2k − 1).
In general, when powers are included, one cannot divide by n.
11 / 48
The conjugacy growth series in free groups
• Rivin (2000, 2010): the conjugacy growth series of Fk is not rational:
σ(z) =
∫ z
0
H(t)
(1− x2)2 + ∞∑ d=1
Theorem (C. - Hermiller, 2012)
For A, B finite groups with generating sets XA = A \ 1A, XB = B \ 1B ,
and A ∗ B with generating set X = XA ∪ XB .
Then σ(A ∗ B,X ) is rational iff A = B = Z/2Z, i.e. A ∗ B = D∞.
13 / 48
Theorem (C. - Hermiller, 2012)
For A, B finite groups with generating sets XA = A \ 1A, XB = B \ 1B ,
and A ∗ B with generating set X = XA ∪ XB .
Then σ(A ∗ B,X ) is rational iff A = B = Z/2Z, i.e. A ∗ B = D∞.
13 / 48
A generating function f (z) is
I rational if there exist polynomials P(z), Q(z) with integer coefficients
such that f (z) = P(z) Q(z)
;
I algebraic if there exists a polynomial P(x , y) with integer coefficients such
that P(z , f (z)) = 0;
I transcendental otherwise.
Rivin’s conjecture ⇒
If G is non-elementary hyperbolic, then the conjugacy growth series σ is not
rational.
Proof. (Antoln-C., 2015)
• Recall: σ(n) := ]{[g ] ∈ G | |g |c = n} is the strict conjugacy growth.
• Let φ(n) := ]{[g ] ∈ G | |g |c ≤ n} be the cumulative conjugacy growth.
15 / 48
Rivin’s conjecture ⇒
If G is non-elementary hyperbolic, then the conjugacy growth series σ is not
rational.
Proof. (Antoln-C., 2015)
• Recall: σ(n) := ]{[g ] ∈ G | |g |c = n} is the strict conjugacy growth.
• Let φ(n) := ]{[g ] ∈ G | |g |c ≤ n} be the cumulative conjugacy growth.
15 / 48
Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper).
Let G be a non-elementary word hyperbolic group. Then there are positive
constants A,B and n0 such that
A ehn
ehn
n
for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|.
MESSAGE:.
The number of conjugacy classes in the ball of radius n is asymptotically the
number of elements in the ball of radius n divided by n.
16 / 48
Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper).
Let G be a non-elementary word hyperbolic group. Then there are positive
constants A,B and n0 such that
A ehn
ehn
n
for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|.
MESSAGE:.
The number of conjugacy classes in the ball of radius n is asymptotically the
number of elements in the ball of radius n divided by n.
16 / 48
Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper).
Let G be a non-elementary word hyperbolic group. Then there are positive
constants A,B and n0 such that
A ehn
ehn
n
for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|.
Lemma (Flajolet: Trancendence of series based on bounds).
Suppose there are positive constants A,B, h and an integer n0 ≥ 0 s.t.
A ehn
i=0 anz n is not algebraic.
17 / 48
Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper).
Let G be a non-elementary word hyperbolic group. Then there are positive
constants A,B and n0 such that
A ehn
ehn
n
for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|.
Lemma (Flajolet: Trancendence of series based on bounds).
Suppose there are positive constants A,B, h and an integer n0 ≥ 0 s.t.
A ehn
i=0 anz n is not algebraic.
17 / 48
Bounds for the conjugacy growth
Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative
conjugacy growth.
Theorem.(Coornaert and Knieper, GAFA 2002)
Let G be a non-elementary word hyperbolic. Then there are positive constants
A and n0 such that for all n ≥ n0
A ehn
n ≤ φp(n).
Theorem.(Coornaert and Knieper, IJAC 2004)
Let G be a torsion-free non-elementary word hyperbolic group. Then there are
positive constants B and n1 such that for all n ≥ n1
φp(n) ≤ B ehn
Bounds for the conjugacy growth
Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative
conjugacy growth.
Theorem.(Coornaert and Knieper, GAFA 2002)
Let G be a non-elementary word hyperbolic. Then there are positive constants
A and n0 such that for all n ≥ n0
A ehn
n ≤ φp(n).
Theorem.(Coornaert and Knieper, IJAC 2004)
Let G be a torsion-free non-elementary word hyperbolic group. Then there are
positive constants B and n1 such that for all n ≥ n1
φp(n) ≤ B ehn
Bounds for the conjugacy growth
Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative
conjugacy growth.
Theorem.(Coornaert and Knieper, GAFA 2002)
Let G be a non-elementary word hyperbolic. Then there are positive constants
A and n0 such that for all n ≥ n0
A ehn
n ≤ φp(n).
Theorem.(Coornaert and Knieper, IJAC 2004)
Let G be a torsion-free non-elementary word hyperbolic group. Then there are
positive constants B and n1 such that for all n ≥ n1
φp(n) ≤ B ehn
Rivin’s conjecture ⇒ : Proof
1. Drop torsion requirement from upper bound of Coornaert and Knieper:
(i) use the fact that there exists m <∞ such that all finite subgroups F ≤ G
satisfy |F | ≤ m.
) cyclic permutations of a primitive conjugacy representative of
length n correspond to different elements of length n in G .
2. Find conjugacy growth upper bound for all conjugacy classes, i.e. include
the non-primitive classes in the count.

Rivin’s conjecture ⇒ : Proof
1. Drop torsion requirement from upper bound of Coornaert and Knieper:
(i) use the fact that there exists m <∞ such that all finite subgroups F ≤ G
satisfy |F | ≤ m.
) cyclic permutations of a primitive conjugacy representative of
length n correspond to different elements of length n in G .
2. Find conjugacy growth upper bound for all conjugacy classes, i.e. include
the non-primitive classes in the count.

1. Rivin’s conjecture for relatively hyperbolic groups?
(a) we need sharp bounds for the standard growth function [Yang] √
(b) we need sharp bounds for the conjugacy growth function.
2. Rivin’s conjecture for acylindrically hyperbolic groups:
Is the conjugacy growth series of a f.g. acylindrically hyperbolic group
transcendental?
1. Rivin’s conjecture for relatively hyperbolic groups?
(a) we need sharp bounds for the standard growth function [Yang] √
(b) we need sharp bounds for the conjugacy growth function.
2. Rivin’s conjecture for acylindrically hyperbolic groups:
Is the conjugacy growth series of a f.g. acylindrically hyperbolic group
transcendental?
1. Rivin’s conjecture for relatively hyperbolic groups?
(a) we need sharp bounds for the standard growth function [Yang] √
(b) we need sharp bounds for the conjugacy growth function.
2. Rivin’s conjecture for acylindrically hyperbolic groups:
Is the conjugacy growth series of a f.g. acylindrically hyperbolic group
transcendental?
21 / 48
Formal languages and the Chomsky hierarchy
Let X be a finite alphabet. A formal language over X is a set L ⊂ X∗ of words.
regular
unambiguous
context-free
context-free
context-sensitive
Formal languages and the Chomsky hierarchy
Let X be a finite alphabet. A formal language over X is a set L ⊂ X∗ of words.
regular
unambiguous
context-free
context-free
context-sensitive
I The growth function fL : N→ N of L is:
fL(n) = ]{w ∈ L | w of length n}.
I The growth series of L is
SL(z) = ∞∑ n=0
I Unambiguous context-free languages have ALGEBRAIC growth series.
(Chomsky-Schutzenberger)
I The growth function fL : N→ N of L is:
fL(n) = ]{w ∈ L | w of length n}.
I The growth series of L is
SL(z) = ∞∑ n=0
I Unambiguous context-free languages have ALGEBRAIC growth series.
(Chomsky-Schutzenberger)
I The growth function fL : N→ N of L is:
fL(n) = ]{w ∈ L | w of length n}.
I The growth series of L is
SL(z) = ∞∑ n=0
I Unambiguous context-free languages have ALGEBRAIC growth series.
(Chomsky-Schutzenberger)
Corollary. [AC]
Let G be a non-elementary hyperbolic group, X a finite generating set and Lc
any set of minimal length representatives of conjugacy classes.
Then Lc is not regular.
By Chomsky-Schuzenberger, Lc is not unambiguous context-free (UCF).
24 / 48
Corollary. [AC]
Let G be a non-elementary hyperbolic group, X a finite generating set and Lc
any set of minimal length representatives of conjugacy classes.
Then Lc is not regular.
By Chomsky-Schuzenberger, Lc is not unambiguous context-free (UCF).
24 / 48
Main Theorem [AC, 2015]
Let G be an acylindrically hyperbolic group, X any finite generating set, and Lc
be a set containing one minimal length representative of each conjugacy class.
Then Lc is not unambiguous context-free, so not regular.
25 / 48
Main Theorem [AC, 2015]
Let G be an acylindrically hyperbolic group, X any finite generating set, and
Lc be a set containing one minimal length representative of each primitive
conjugacy class/commensurating class.
26 / 48
(Dahmani, Guirardel, Osin, Hamenstadt, Bowditch, Fujiwara, Minasyan ...)
I relatively…

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